Josephson junctions have been realized in which two superconducting magnesium-diboride (MgB2) layers are separated by a thin MgO barrier layer, using the ramp-type configuration. Their current–voltage characteristics follow the behavior described by the resistively shunted junction model, with an excess current of about 30% of the critical current Ic. A suppression of 70% of Ic was achieved in applied magnetic fields. Shapiro steps were observed by irradiating the junctions with 10.0 GHz microwaves, and the dependence of the step height on applied rf current is well described by a current–source model. Reference samples prepared without the MgO layer showed strong-link behavior with large Ic values.